A Class of Linear-Quadratic-Gaussian (LQG) Mean-Field Game (MFG) of Stochastic Delay Systems

TL;DR

This paper develops a novel framework for analyzing linear-quadratic-Gaussian mean-field games involving stochastic delay systems, establishing well-posedness and decentralized strategies with epsilon-Nash equilibrium properties.

Contribution

It introduces a new approach using anticipated forward-backward stochastic differential delay equations for the consistency condition, avoiding classical fixed-point analysis.

Findings

Well-posedness of the consistency system established via continuation method.

Decentralized strategies satisfying epsilon-Nash equilibrium derived.

Analysis of two special cases of delayed mean-field games.

Abstract

This paper investigates the linear-quadratic-Gaussian (LQG) mean-field game (MFG) for a class of stochastic delay systems. We consider a large population system in which the dynamics of each player satisfies some forward stochastic differential delay equation (SDDE). The consistency condition or Nash certainty equivalence (NCE) principle is established through an auxiliary mean-field system of anticipated forward-backward stochastic differential equation with delay (AFBSDDE). The wellposedness of such consistency condition system can be further established by some continuation method instead the classical fixed-point analysis. Thus, the consistency condition maybe given on arbitrary time horizon. The decentralized strategies are derived which are shown to satisfy the $\epsilon$-Nash equilibrium property. Two special cases of our MFG for delayed system are further investigated.